Ordinary fractions, theory and operations: adding, subtracting, multiplying, dividing, reducing (simplifying), comparing, sorting fractions

Ordinary fractions: what are they?

If we evenly divide six apples to 3 children, then we divide 6 by 3 = 2 and we know that every child will get 2 apples. If we have to share 2 apples to 3 children, then the partition of 2 by 3 has to be solved.

This operation, 2:3, has no solution in the set of the natural numbers. However, we can still divide the apples with a knife. The amount of the apple for each child will be defined by using the fraction 2/3. All the similar cases lead to fractions.

Fractions are formed by dividing one or more integers:

These fractions in which both the numerator and denominator are integers are called simple fractions, common fractions, vulgar fractions or ordinary fractions.

Fractions' numerators and denominators can also be negative, ie: -3/5, -2/-9 , 7/-4; due to sign rule: -3/5 = 3/-5 = -(3/5); -2/-9 = 2/9; 7/-4 = -7/4 = -(7/4)

Ordinary fractions types:

• Proper fractions: 2/3, 1/7, 5/9, -11/13, 10/11, 15/-16; the absolute value (see below, Note 1) of the numerator is smaller than the absolute value of the denominator, then the absolute value of the fraction is smaller than 1.
• Improper or top-heavy fractions: 4/3, 16/3, 9/8, 123/-13; the absolute value of the numerator is larger than or equal to the absolute value of the denominator; the absolute value of the fraction is larger than or equal to 1.
• Improper fractions cand be written as mixed numbers (also called mixed fractions):
• 4/3 = 3/3 + 1/3 = 1 + 1/3, written as: 1 1/3
• 16/3 = 15/3 + 1/3 = 5 + 1/3, written as: 5 1/3
• 9/8 = 8/8 + 1/8 = 1 + 1/8, written as: 1 1/8
• 123/-13 = -123/13 = - (117 + 6) / 13 = - 117/13 - 6/13 = - 9 - 6/13, written as: -9 6/13

(Note 1) Absolute value = The numerical value of a number without regard to its sign. For example, the absolute value of -7 (written │-7│) is 7. More exemples: |-17| = 17; |10| = 10; |-123| = 123; ...

• If the numerator of a fraction is equal to the denominator of another fraction and vice versa, then the fractions are called reciprocal:
• 3/5 and 5/3
• 17/6 and 6/17
• The product of a fraction and its reciprocal is 1, the reciprocal is the multiplicative inverse of a fraction.

How do fractions compare?

• If two positive fractions have the same denominator, then the larger the fraction's numerator the larger the fraction: 2/7 < 6/7.
• If two positive fractions have the same numerator, then the larger the fraction's denominator the smaller the fraction: 5/9 < 5/7
• If two positive fractions have different numerators and denominators, first the fractions are brought to the same denominator, and the fraction with larger numerator is the larger: 8/9 vs. 5/7 => (8 * 7) / (9 * 7) vs. (5 * 9) / (7 * 9) => 56/63 > 45/63 => 8/9 > 5/7
• If two negative fractions have the same denominator, then the larger the fraction's numerator the smaller the fraction: -2/7 >; -6/7.
• If two negative fractions have the same numerator, then the larger the fraction's denominator the larger the fraction: -5/9 > -5/7
• If two negative fractions have different numerators and denominators, first the fractions are brought to the same denominator, and the fraction with larger numerator is the smaller one: -8/9 vs. -5/7 => - (8 * 7) / (9 * 7) vs. - (5 * 9) / (7 * 9) => - 56/63 < - 45/63 => -8/9 < -5/7